Questions are taken verbatim from the NCERT textbook; answers were grounded against the chapter's content during generation. Items needing review are marked.
Your Progress - Chapter 60% complete
1Section 6.1 - Perimeter6 questions
Q.1Find the missing terms:
a. Perimeter of a rectangle = 14 cm; breadth = 2 cm; length = ?.
b. Perimeter of a square = 20 cm; side of a length = ?.
c. Perimeter of a rectangle = 12 m; length = 3 m; breadth = ?.v
Solutiona. $2(l+2)=14$, so $l+2=7$ and $l=5\text{ cm}$. b. For a square, $4s=20$, so $s=5\text{ cm}$. c. $2(3+b)=12$, so $3+b=6$ and $b=3\text{ m}$.
Answer:a. $5\text{ cm}$; b. $5\text{ cm}$; c. $3\text{ m}$.
Q.2A rectangle having sidelengths 5 cm and 3 cm is made using a piece of wire. If the wire is straightened and then bent to form a square, what will be the length of a side of the square?v
SolutionThe wire length is the rectangle perimeter: $2(5+3)=16\text{ cm}$. A square made from the same wire has side $16\div4=4\text{ cm}$.
Q.3Find the length of the third side of a triangle having a perimeter of 55 cm and having two sides of length 20 cm and 14 cm, respectively.v
SolutionThird side $=55-20-14=21\text{ cm}$.
Q.4What would be the cost of fencing a rectangular park whose length is 150 m and breadth is 120 m, if the fence costs ₹40 per metre?v
SolutionPerimeter $=2(150+120)=540\text{ m}$. Cost $=540\times40=21600$, so the fencing costs $Rs\ 21,600$.
Q.5A piece of string is 36 cm long. What will be the length of each side, if it is used to form:
a. A square,
b. A triangle with all sides of equal length, and
c. A hexagon (a six sided closed figure) with sides of equal length?v
SolutionDivide the total string length by the number of equal sides: square $36\div4=9$, triangle $36\div3=12$, hexagon $36\div6=6$.
Answer:a. $9\text{ cm}$; b. $12\text{ cm}$; c. $6\text{ cm}$.
Q.6A farmer has a rectangular field having length 230 m and breadth 160 m. He wants to fence it with 3 rounds of rope as shown. What is the total length of rope needed?v
SolutionOne round is the perimeter: $2(230+160)=780\text{ m}$. Three rounds require $3\times780=2340\text{ m}$.
2Section 6.1 - Running Tracks2 questions
Q.1Find out the total distance Akshi has covered in 5 rounds.v
SolutionAkshi's outer track perimeter is $2(70+40)=220\text{ m}$. In $5$ rounds she covers $5\times220=1100\text{ m}$.
Q.2Find out the total distance Toshi has covered in 7 rounds. Who ran a longer distance?v
SolutionToshi's inner track perimeter is $2(60+30)=180\text{ m}$. In $7$ rounds she covers $7\times180=1260\text{ m}$, which is more than Akshi's $1100\text{ m}$.
Answer:Toshi covered $1260\text{ m}$ and ran the longer distance.
3Section 6.1 - Diagonal Units1 questions
Q.1Akshi says that the perimeter of this triangle shape is 9 units. Toshi says it can’t be 9 units and the perimeter will be more than 9 units. What do you think?v
SolutionA diagonal of a square is longer than its side. If diagonal boundary pieces are counted as one side unit, the perimeter is underestimated.
Answer:Toshi is correct; the perimeter is more than $9$ units.
4Section 6.1 - Regular Polygons1 questions
Q.1Find various objects from your surroundings that have regular shapes and find their perimeters. Also, generalise your understanding for the perimeter of other regular polygons.v
SolutionBecause all sides of a regular polygon are equal, adding all side lengths is the same as multiplying one side length by the number of sides.
Answer:For a regular polygon, perimeter $=$ number of sides $\times$ length of one side.
5Section 6.2 - Area3 questions
Q.1The area of a rectangular garden 25 m long is 300 sq m. What is the width of the garden?v
SolutionWidth $=$ area $\div$ length $=300\div25=12\text{ m}$.
Q.2What is the cost of tiling a rectangular plot of land 500 m long and 200 m wide at the rate of ₹8 per hundred sq m?v
SolutionArea $=500\times200=100000\text{ sq m}$. There are $100000\div100=1000$ hundreds of square metres. Cost $=1000\times8=Rs\ 8000$.
Q.3A rectangular coconut grove is 100 m long and 50 m wide. If each coconut tree requires 25 sq m, what is the maximum number of trees that can be planted in this grove?v
SolutionGrove area $=100\times50=5000\text{ sq m}$. Number of trees $=5000\div25=200$.
6Section 6.2 - Tangram8 questions
Q.1Explore and figure out how many pieces have the same area.v
SolutionUsing Shape C as one unit of area, the areas are A = B = $4C$, C = E = $C$, and D = F = G = $2C$.
Answer:Shapes A and B have the same area; Shapes C and E have the same area; Shapes D, F and G have the same area.
Q.2How many times bigger is Shape D as compared to Shape C? What is the relationship between Shapes C, D and E?v
SolutionThe hint states that Shape D can be covered exactly using Shapes C and E, and C and E have equal area. Thus $D=C+E=2C$.
Answer:Shape D is twice Shape C. Shapes C and E have the same area, and together they exactly cover Shape D.
Q.3Which shape has more area: Shape D or F? Give reasons for your answer.v
SolutionBoth Shape D and Shape F have area equal to $2$ times Shape C in the tangram.
Answer:They have the same area.
Q.4Which shape has more area: Shape F or G? Give reasons for your answer.v
SolutionBoth Shape F and Shape G have area equal to $2$ times Shape C.
Answer:They have the same area.
Q.5What is the area of Shape A as compared to Shape G? Is it twice as big? Four times as big?v
SolutionIn terms of Shape C, $A=4C$ and $G=2C$, so $A=2G$.
Answer:Shape A is twice as big as Shape G.
Q.6Can you now figure out the area of the big square formed with all seven pieces in terms of the area of Shape C?v
SolutionAdd all pieces in units of Shape C: $A+B+C+D+E+F+G=4+4+1+2+1+2+2=16$.
Answer:$16$ times the area of Shape C.
Q.7Arrange these 7 pieces to form a rectangle. What will be the area of this rectangle in terms of the area of Shape C now? Give reasons for your answer.v
SolutionRearranging the same seven pieces does not change their total area. Therefore the rectangle has the same area as the big square, $16C$.
Answer:$16$ times the area of Shape C.
Q.8Are the perimeters of the square and the rectangle formed from these 7 pieces different or the same? Give an explanation for your answer.v
SolutionThe same pieces keep the same total area when rearranged, but the outside boundary can change. Therefore the perimeter of the square and the perimeter of a rectangle made from the same pieces need not be the same.
Answer:They can be different.
7Section 6.3 - Figure it Out8 questions
Q.1Give the dimensions of a rectangle whose area is the sum of the areas of these two rectangles having measurements: 5 m × 10 m and 2 m × 7 m.v
SolutionThe total area is $5\times10+2\times7=50+14=64\text{ sq m}$. Any rectangle with area $64\text{ sq m}$ works.
Answer:Examples: $16\text{ m}\times4\text{ m}$, $32\text{ m}\times2\text{ m}$, or $8\text{ m}\times8\text{ m}$.
Q.2The area of a rectangular garden that is 50 m long is 1000 sq m. Find the width of the garden.v
SolutionWidth $=$ area $\div$ length $=1000\div50=20\text{ m}$.
Q.3The floor of a room is 5 m long and 4 m wide. A square carpet whose sides are 3 m in length is laid on the floor. Find the area that is not carpeted.v
SolutionFloor area $=5\times4=20\text{ sq m}$. Carpet area $=3\times3=9\text{ sq m}$. Uncarpeted area $=20-9=11\text{ sq m}$.
Q.4Four flower beds having sides 2 m long and 1 m wide are dug at the four corners of a garden that is 15 m long and 12 m wide. How much area is now available for laying down a lawn?v
SolutionGarden area $=15\times12=180\text{ sq m}$. Each flower bed has area $2\times1=2\text{ sq m}$, so four beds occupy $8\text{ sq m}$. Lawn area $=180-8=172\text{ sq m}$.
Answer:$172\text{ sq m}$.
Q.5Shape A has an area of 18 square units and Shape B has an area of 20 square units. Shape A has a longer perimeter than Shape B. Draw two such shapes satisfying the given conditions.v
SolutionShape A area $=2\times9=18$ and perimeter $=2(2+9)=22$. Shape B area $=4\times5=20$ and perimeter $=2(4+5)=18$. Thus Shape A has a smaller area but a longer perimeter.
Answer:One possible choice is Shape A as a $2\times9$ rectangle and Shape B as a $4\times5$ rectangle.
Q.6On a page in your book, draw a rectangular border that is 1 cm from the top and bottom and 1.5 cm from the left and right sides. What is the perimeter of the border?v
SolutionIf the page length is $L\text{ cm}$ and width is $W\text{ cm}$, the border rectangle has dimensions $(L-2)\text{ cm}$ by $(W-3)\text{ cm}$. Its perimeter is $2[(L-2)+(W-3)]=2L+2W-10\text{ cm}$.
Answer:The answer depends on the page dimensions.
Q.7Draw a rectangle of size 12 units × 8 units. Draw another rectangle inside it, without touching the outer rectangle that occupies exactly half the area.v
SolutionThe outer rectangle area is $12\times8=96$ square units. Half of this is $96\div2=48$ square units. Any inner rectangle not touching the boundary and having area $48$ square units satisfies the condition.
Answer:The inner rectangle should have area $48$ square units.
Q.8A square piece of paper is folded in half. The square is then cut into two rectangles along the fold. Regardless of the size of the square, one of the following statements is always true. Which statement is true here?
a. The area of each rectangle is larger than the area of the square.
b. The perimeter of the square is greater than the perimeters of both the rectangles added together.
c. The perimeters of both the rectangles added together is always 1 1/2 times the perimeter of the square.
d. The area of the square is always three times as large as the areas of both rectangles added together.v
- a. The area of each rectangle is larger than the area of the square.
- b. The perimeter of the square is greater than the perimeters of both the rectangles added together.
- c. The perimeters of both the rectangles added together is always 1 1/2 times the perimeter of the square.
- d. The area of the square is always three times as large as the areas of both rectangles added together.
SolutionLet the square side be $s$. Each rectangle is $s\times\frac{s}{2}$, so each rectangle has perimeter $2(s+\frac{s}{2})=3s$. The two rectangle perimeters add to $6s$. The square perimeter is $4s$, and $6s=1\frac{1}{2}\times4s$.